| L(s) = 1 | − 3-s − 5·5-s + 3·7-s − 9-s + 3·11-s + 4·13-s + 5·15-s + 6·17-s − 3·19-s − 3·21-s + 2·23-s + 7·25-s − 7·27-s − 5·29-s − 4·31-s − 3·33-s − 15·35-s − 7·37-s − 4·39-s − 7·41-s − 43-s + 5·45-s + 11·47-s + 6·49-s − 6·51-s − 3·53-s − 15·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2.23·5-s + 1.13·7-s − 1/3·9-s + 0.904·11-s + 1.10·13-s + 1.29·15-s + 1.45·17-s − 0.688·19-s − 0.654·21-s + 0.417·23-s + 7/5·25-s − 1.34·27-s − 0.928·29-s − 0.718·31-s − 0.522·33-s − 2.53·35-s − 1.15·37-s − 0.640·39-s − 1.09·41-s − 0.152·43-s + 0.745·45-s + 1.60·47-s + 6/7·49-s − 0.840·51-s − 0.412·53-s − 2.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 2 T^{2} + 10 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + p T + 18 T^{2} + 48 T^{3} + 18 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 8 T^{2} + 10 T^{3} + 8 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 23 T^{2} - 96 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 11 T^{2} + 20 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 49 T^{2} - 60 T^{3} + 49 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 5 T + 60 T^{2} + 252 T^{3} + 60 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 184 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 7 T + 92 T^{2} + 432 T^{3} + 92 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 7 T + 134 T^{2} + 572 T^{3} + 134 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + T + 94 T^{2} + 58 T^{3} + 94 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 11 T + 174 T^{2} - 1050 T^{3} + 174 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 3 T + 96 T^{2} + 80 T^{3} + 96 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 3 T + 132 T^{2} + 246 T^{3} + 132 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 7 T + 170 T^{2} + 852 T^{3} + 170 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 197 T^{2} - 1592 T^{3} + 197 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 9 T + 212 T^{2} + 1270 T^{3} + 212 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 107 T^{2} - 392 T^{3} + 107 p T^{4} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 15 T + 278 T^{2} + 2386 T^{3} + 278 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 16 T + 3 p T^{2} + 2208 T^{3} + 3 p^{2} T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 3 T + 242 T^{2} + 556 T^{3} + 242 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 5 T + 270 T^{2} + 872 T^{3} + 270 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.11112182115085410648606980086, −6.95771083105017229074348701386, −6.91134600153018472074665403871, −6.82815965780422193161091147174, −6.10288505403354802736893225828, −5.96052398828219944581829884458, −5.87926827515724052261598016820, −5.52001643745958183784302632548, −5.35977906315062855585023326234, −5.35371292017920075780715973889, −4.75295064691961968010898748900, −4.48925849427397303278459854767, −4.41228469564613108722655616541, −4.00095373329848451302148825250, −3.92025522831354794269075256813, −3.68222479185258329361392215653, −3.49561870694169322802113584692, −3.29812113484374428356532447535, −2.95908749824610114709844453550, −2.43984940371804617205461236204, −2.24055631838275322536401146455, −1.64258285855059893799413357882, −1.57283399695533435773204712778, −1.14871470525869076832307503918, −1.04303462841050887097293724808, 0, 0, 0,
1.04303462841050887097293724808, 1.14871470525869076832307503918, 1.57283399695533435773204712778, 1.64258285855059893799413357882, 2.24055631838275322536401146455, 2.43984940371804617205461236204, 2.95908749824610114709844453550, 3.29812113484374428356532447535, 3.49561870694169322802113584692, 3.68222479185258329361392215653, 3.92025522831354794269075256813, 4.00095373329848451302148825250, 4.41228469564613108722655616541, 4.48925849427397303278459854767, 4.75295064691961968010898748900, 5.35371292017920075780715973889, 5.35977906315062855585023326234, 5.52001643745958183784302632548, 5.87926827515724052261598016820, 5.96052398828219944581829884458, 6.10288505403354802736893225828, 6.82815965780422193161091147174, 6.91134600153018472074665403871, 6.95771083105017229074348701386, 7.11112182115085410648606980086
